§ Motivation for modal logic
possibly A -> necessarily (possibly A -> B) -> necessarily B
- this weakens the precondition
A -> (A -> B) -> B by needing only
possible Aand strengthens the postcondition by spitting out
- Key idea 1: if A is true in no world
possibly A that we have is false, and from this we derive explosion.
- Key idea 2: if A is true in some world
wa, then suppose we are in some arbitrary world
A is true in
wa, we have
necessarily (possibly A -> B) is true in all worlds, we have
(possibly A -> B).
- Since we have both
possibly A, and
possibly A -> B, we derive
wr was arbitrary, we then have
necessarily B since
B holds in any arbitrary worlds.
§ Use of this for Kant
- experience of objects is possible.
- it is necessarily the case that if experience is possible, then I must have some way to unite experience.
- thus, necessarily we have unity of experience.
§ Use of this for descartes
- it is possible for me to be certain of something (ie, I think therefore I am)
- it is neecessarily the case that if I can be certain of something, I have clear and distinct perception.
- Therefore, it is necessary that I have clear and distinct perception.